# ag.algebraic geometry – Chiral homology for the Virasoro algebra and/or affine Lie algebra

I want to understand what concrete analytical objects are found in chiral homology of higher degree of a vertex algera (-module) $$M$$. More precisely: I can obtain conformal blocks on a surface $$Sigma$$ for the Virasoro algebra as invariants of $$M$$ under the action of vector fields – for the torus one can then indeed work out a modular function depending on the structure of the torus and possibly the location of singularities. One can also compute elements in it by gluing the vertex operator on a 3-punctured sphere to a torus (a.k.a graded character of $$M$$)

But what about higher homologies: In the Lie algebra standard complex the chains are functions depending on an $$n$$-tuple of (here) vector fields up to…how does this relate for the torus to modular functions or something similar?

Same question for affine Lie algebra at negative level – I have found the great work of Gaitsgory, but I would like to know if there is any concrete analytic realization of the elements by…?

(But maybe it is simply the wrong question !?)

Also, I would already be very happy for reference like to as for references you might have on the chiral cohomology of the Virasoro algebra on surfaces (I know the computation for $$M$$ trivial, but for say $$M$$ a irrep of a minimal model?).